 19.19.1: In each of 1 through 10, carry out the indicated calculation.(3 4i)...
 19.19.35: In each of 1 through 12, find u and v so that f = u + iv, determine...
 19.19.48: In each of 1 through 10, write the function value in the form a + b...
 19.19.66: In each of 1 through 6, determine all values of the complex logarit...
 19.19.74: In each of 1 through 14, determine all values of zw.i 1+i
 19.19.2: In each of 1 through 10, carry out the indicated calculation.i(6 2i...
 19.19.36: In each of 1 through 12, find u and v so that f = u + iv, determine...
 19.19.49: In each of 1 through 10, write the function value in the form a + b...
 19.19.67: In each of 1 through 6, determine all values of the complex logarit...
 19.19.75: In each of 1 through 14, determine all values of zw.(1 + i)2
 19.19.3: In each of 1 through 10, carry out the indicated calculation.(2 + i...
 19.19.37: In each of 1 through 12, find u and v so that f = u + iv, determine...
 19.19.50: In each of 1 through 10, write the function value in the form a + b...
 19.19.68: In each of 1 through 6, determine all values of the complex logarit...
 19.19.76: In each of 1 through 14, determine all values of zw.ii
 19.19.4: In each of 1 through 10, carry out the indicated calculation.((2 + ...
 19.19.38: In each of 1 through 12, find u and v so that f = u + iv, determine...
 19.19.51: In each of 1 through 10, write the function value in the form a + b...
 19.19.69: In each of 1 through 6, determine all values of the complex logarit...
 19.19.77: In each of 1 through 14, determine all values of zw.. (1 + i)2i
 19.19.5: In each of 1 through 10, carry out the indicated calculation.(17 6i...
 19.19.39: In each of 1 through 12, find u and v so that f = u + iv, determine...
 19.19.52: In each of 1 through 10, write the function value in the form a + b...
 19.19.70: In each of 1 through 6, determine all values of the complex logarit...
 19.19.78: In each of 1 through 14, determine all values of zw.(1 + i)3i
 19.19.6: In each of 1 through 10, carry out the indicated calculation.3i/(4...
 19.19.40: In each of 1 through 12, find u and v so that f = u + iv, determine...
 19.19.53: In each of 1 through 10, write the function value in the form a + b...
 19.19.71: In each of 1 through 6, determine all values of the complex logarit...
 19.19.79: In each of 1 through 14, determine all values of zw.(1 i)1/3
 19.19.7: In each of 1 through 10, carry out the indicated calculation.i 3 4i...
 19.19.41: In each of 1 through 12, find u and v so that f = u + iv, determine...
 19.19.54: In each of 1 through 10, write the function value in the form a + b...
 19.19.72: Let z and w be nonzero complex numbers. Show that each value of log...
 19.19.80: In each of 1 through 14, determine all values of zw.i 1/4
 19.19.8: In each of 1 through 10, carry out the indicated calculation.(3 + i)3
 19.19.42: In each of 1 through 12, find u and v so that f = u + iv, determine...
 19.19.55: In each of 1 through 10, write the function value in the form a + b...
 19.19.73: Let z and w be nonzero complex numbers. Show that each value of log...
 19.19.81: In each of 1 through 14, determine all values of zw.161/4
 19.19.9: In each of 1 through 10, carry out the indicated calculation.((6 + ...
 19.19.43: In each of 1 through 12, find u and v so that f = u + iv, determine...
 19.19.56: In each of 1 through 10, write the function value in the form a + b...
 19.19.82: In each of 1 through 14, determine all values of zw.(4)2i
 19.19.10: In each of 1 through 10, carry out the indicated calculation.(3 8i)...
 19.19.44: In each of 1 through 12, find u and v so that f = u + iv, determine...
 19.19.57: In each of 1 through 10, write the function value in the form a + b...
 19.19.83: In each of 1 through 14, determine all values of zw.623i
 19.19.11: In each of 11 through 16, determine the magnitudeand all of the arg...
 19.19.45: In each of 1 through 12, find u and v so that f = u + iv, determine...
 19.19.58: Determine u and v such that ez2 = u(x, y) + iv(x, y). Show that u a...
 19.19.84: In each of 1 through 14, determine all values of zw.(16)1/4
 19.19.12: In each of 11 through 16, determine the magnitude and all of the ar...
 19.19.46: In each of 1 through 12, find u and v so that f = u + iv, determine...
 19.19.59: Find u and v such that e1/z =u(x, y)+iv(x, y). Show that u and v sa...
 19.19.85: In each of 1 through 14, determine all values of zw.][(1 + i)/(1 i)...
 19.19.13: In each of 11 through 16, determine the magnitude and all of the ar...
 19.19.47: Let zn = an + ibn be a sequence of complex numbers. We say that thi...
 19.19.60: Find u and v such that zez = u(x, y) +iv(x, y). Show that u and v s...
 19.19.86: In each of 1 through 14, determine all values of zw.11/6/2017
 19.19.14: In each of 11 through 16, determine the magnitude and all of the ar...
 19.19.61: Find u and v such that cos2 (z) = u(x, y) + iv(x, y). Show that u a...
 19.19.87: In each of 1 through 14, determine all values of zw.(7i)3i
 19.19.15: In each of 11 through 16, determine the magnitude and all of the ar...
 19.19.62: Find all solutions of ez = 2i.
 19.19.88: Let u1, , un be the nth roots of unity with n a positive integer an...
 19.19.16: In each of 11 through 16, determine the magnitude and all of the ar...
 19.19.63: Derive the following identities. (a) sin(z + w) = sin(z) cos(w) + c...
 19.19.89: Let n be a positive integer, and let =e2i/n . Evaluate n1 j=0(1)j j .
 19.19.17: In each of 17 through 22, write the number in polar form2 + 2i
 19.19.64: Find all solutions of ez = 2.
 19.19.18: In each of 17 through 22, write the number in polar form7i
 19.19.65: Find all solutions of sin(z) = i.
 19.19.19: In each of 17 through 22, write the number in polar form5 2i
 19.19.20: In each of 17 through 22, write the number in polar form4 i
 19.19.21: In each of 17 through 22, write the number in polar form8 + i
 19.19.22: In each of 17 through 22, write the number in polar form12 + 3i
 19.19.23: Show that, for any positive integer n, i 4n = 1,i 4n+1 = i,i 4n+2 =...
 19.19.24: Let z = a + ib. Determine Re(z2 ) and Im(z2 )
 19.19.25: Show that complex numbers z, w, and u form vertices of an equilater...
 19.19.26: Show that z2 = (z)2 if and only if z is either real or pure imaginary.
 19.19.27: Let z and w be numbers with zw = 1. Suppose either z or w has magni...
 19.19.28: Show that, for any numbers z and w, z + w 2 + z w 2 = 2 z 2 +...
 19.19.29: In each of 29 through 37, a set of complex numbers is specified. De...
 19.19.30: In each of 29 through 37, a set of complex numbers is specified. De...
 19.19.31: In each of 29 through 37, a set of complex numbers is specified. De...
 19.19.32: In each of 29 through 37, a set of complex numbers is specified. De...
 19.19.33: In each of 29 through 37, a set of complex numbers is specified. De...
 19.19.34: In each of 29 through 37, a set of complex numbers is specified. De...
Solutions for Chapter 19: Complex Numbers and Functions
Full solutions for Advanced Engineering Mathematics  7th Edition
ISBN: 9781111427412
Solutions for Chapter 19: Complex Numbers and Functions
Get Full SolutionsChapter 19: Complex Numbers and Functions includes 89 full stepbystep solutions. Advanced Engineering Mathematics was written by and is associated to the ISBN: 9781111427412. Since 89 problems in chapter 19: Complex Numbers and Functions have been answered, more than 36861 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Advanced Engineering Mathematics, edition: 7.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)ยท(b  Ax) = o.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.